Chapter 2 Rheological Models: Integral and Differential Representations

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1 Chapter 2 Rheological Models: Integral and Differential Representations Viscoelastic relations may be expressed in both integral and differential forms. Integral forms are very general and appropriate for theoretical work. Differential forms are related to rheological models that provide a more direct physical interpretation of viscoelastic behavior. In this chapter we describe the most usual rheological models, deduce their differential equations and, by solving them, we find the corresponding integral representations. These relations will be set in a more computational friendly form in Chap. 3 and extended to three-dimensional situations in Chap. 4 and then used in analytical and computational solutions. 2.1 General Integral Relations When the functional relation (1.5) in Chap. 1 is linear it has a simple and useful representation given by the Riesz theorem [1]: if the functional D is linear and equi-continuous, it can be written or Dðt sþdrðsþ ð2:1þ Dðt sþ _rðsþds ð2:2þ Here, should be chosen in a way that for s \ the material is at rest, without stress and strain. From the relations above we see that D(t s)h(t ) represents S. P. C. Marques and G. J. Creus, Computational Viscoelasticity, SpringerBriefs in Computational Mechanics, DOI: / _2, Ó The Author(s)

2 12 2 Rheological Models: Integral and Differential Representations the strain corresponding to a creep test with r(t) = H(t ). D(t s) is the creep function or creep compliance of dimension L 2 /F. Equation (2.1) is an integral of the Stieltjes type. For these integrals, when r(s) has steps Dr i H(t s i ) we have X Dðt sþdrðsþþ Dr i Dðt s i Þ i ð2:3þ As long as r(t) is continuous and differentiable, _rðtþ exists and the form (2.2) can be used. Notice that the integration is performed with relation to s; t acts as a parameter and as the superior limit of integration, but it is a constant inside the integral. Thus, for differentiation in relation to t we have to use the Leibnitz formula (See Appendix A). Alternative forms of the integral representation. Besides the relations (2.1) and (2.2) we may use the inverse relations and rðtþ ¼ rðtþ ¼ Eðt sþdeðsþ ð2:4þ Eðt sþ_eðsþds ð2:5þ exchanging the roles of stress and strain. E(t - s) is the specific relaxation function, i.e., the stress response to a unit step of strain. Integrating (2.2) by parts, we obtain where rðtþ Eð0Þ þ dðt sþrðsþds ð2:6þ dðt sþ ¼ o Dðt sþ os Eð0Þ ¼1=Dðt tþ ð2:7þ Sometimes, instantaneous and delayed components of the specific creep are separated Dðt sþ ¼ 1 þ Cðt sþ Eð0Þ Eðt sþ ¼Eð1Þ þ Rðt sþ ð2:8þ

3 2.2 Rheological Models 13 Fig. 2.1 a Hooke model (spring). b Newton model (dashpot) 2.2 Rheological Models The behavior of viscoelastic materials under uniaxial loading may be represented by means of conceptual models composed of elastic and viscous elements which provide physical insight and have didactic value. Rheological models are described in most of the books on viscoelasticity such as Flugge [2], Christensen [3] and many others The Basic Elements: Spring and Dashpot An ideal helicoidal spring, perfectly linear elastic and massless, represents Hooke model (see Fig. 2.1a): rðtþ ¼EeðtÞ ð2:9þ where E is the elasticity modulus with dimension [F/L 2 ]. Both length and crosssection are given unit values in order to identify force with stress and elongation with strain. The dashpot (Fig. 2.1b) is an ideal viscous element that extends at a rate proportional to the applied stress, according to Newton equation _rðtþ=g ð2:10þ where _e ¼ oe=ot is the rate of strain and g is the viscosity coefficient, with dimension [FT/L 2 ]. Combining springs and dashpots we obtain different models of viscoelastic behavior. The simplest viscoelastic models are those named after the scientists J. C. Maxwell and Lord Kelvin Maxwell Model This model is the combination of a spring and a dashpot in series, Fig. 2.2a. For this system we may write the equations

4 14 2 Rheological Models: Integral and Differential Representations Fig. 2.2 Maxwell model: a rheological model, b creep test, c relaxation test e E ðtþþe g ðtþ r E ðtþ ¼r g ðtþ ¼rðtÞ r E ðtþ ¼Ee E ðtþ; r g ðtþ ¼g_e g ðtþ ð2:11þ where the sub-indexes g and E indicate dashpot and spring respectively. Differentiating the first Eq with respect to time t and using the constitutive relations for both spring and dashpot, we obtain rðtþ E þ rðtþ g rðtþ ¼0 for t\ ð2:12þ which is the differential equation for the Maxwell model. Solutions of (2.12) may be determined considering either stress or strain as the controlled variable. In the first case we have directly rðtþ E þ 1 rðsþds ð2:13þ g Integrating (2.13) by parts we obtain the alternative expression Comparing this to (2.2) we see that 1 E þ t s _rðsþds g ð2:14þ Dðt sþ ¼ 1 E þ t s g ; t s ð2:15þ is the creep function. Since the strain response is unbounded for t??, one says that the Maxwell model exhibits unbounded creep and sometimes refers to it as Maxwell fluid. For a stress history r(t) = r 0 [H(t ) - H(t s 1 )], with

5 2.2 Rheological Models 15 Fig. 2.3 Kelvin model: a creep test, b rheological model, c relaxation test \ s 1, as that shown in Fig. 2.2b, a residual deformation remains after unloading. Considering now the strain history as given, we obtain from (2.12), using the general solution for first order differential equations in Appendix A, rðtþ ¼E e E g ðt sþ _eðsþds ð2:16þ Then, comparing to (2.5) we see that Eðt Þ¼Ee ðt Þ=T ; T ¼ g E ; t ð2:17þ is the relaxation function that vanishes for t??. Relations using creep or relaxation functions, such as (2.14) and (2.16), are equivalent. A procedure to obtain one from the other is given in Sect. 2.4 and in Chap. 5. The constant T = g/e that appears in the exponential in (2.17) determines the rate of the relaxation process and is called relaxation time. The smaller the relaxation time, the faster the relaxation process, even though total relaxation takes theoretically an infinite time. For example, for t = 3T about 95% of the total relaxation is completed. Considering a loading unloading history, such as e(t) = e 0 [H(t - ) - H(t - s 1 )], with \ s 1, the stress response changes signal (Fig. 2.2c) Kelvin Model This model combines a spring and a dashpot in parallel, Fig. 2.3b. From the relations

6 16 2 Rheological Models: Integral and Differential Representations rðtþ ¼r E ðtþþr g ðtþ e E ðtþ ¼e g ðtþ ¼eðtÞ r E ðtþ ¼Ee E ðtþ; r g ðtþ ¼g_eðtÞ ð2:18þ we can determine the differential equation rðtþ ¼EeðtÞþg_eðtÞ ð2:19þ For a given strain history we have the stress directly from (2.19). A relaxation test is physically impossible with the Kelvin model because _e 0 dðtþ and the corresponding initial stress should be infinitely high. For a given stress history r(t) the solution of (2.19) is 1 g rðsþe t s h ds ; h ¼ g E ð2:20þ Comparing to (2.2) we see that Dðt Þ¼ 1 E 1 e ðt Þ=h ; t ð2:21þ is the creep function for the Kelvin model. For t?? we obtain e(?) = r 0 /E that corresponds to the asymptotic elastic solution, when all the stress is carried by the spring. Again, we have equivalent differential and integral representations. Fig. 2.3 shows the results of creep and relaxation tests. The constant h is called retardation time and is analogous in meaning to the relaxation time: an estimate of the time required for the creep process to approach completion. 2.3 Generalized Models Maxwell and Kelvin models are adequate for qualitative and conceptual analyses, but generally poor for the quantitative representation of the behavior of real materials. In order to improve the representation we need to increase the number of parameters by combining a number of springs and dashpots. A systematic way to do that is to build generalized Maxwell and Kelvin models, shown in Fig The generalized Maxwell model is composed of n? 1 constituent elements in parallel, being n Maxwell models and an isolated spring (to warrant solid behavior) (see Fig. 2.4a). The differential Eq for a generic Maxwell element r of a generalized Maxwell model may be written in the operational form

7 2.3 Generalized Models 17 Fig. 2.4 Maxwell and Kelvin chains with instantaneous elasticity o ot 1 o E r ot þ 1 r r g r ð2:22þ where E r, g r and r r indicate the elastic constant, viscosity coefficients and stress of the r-th element, respectively. The symbol q/qt is a differential operator that can be handled as an algebraic entity. For the generalized Maxwell model the strain is the same for all constituent elements and the total stress is given by the equation 0 1 o=ot rðtþ ¼@ E 1 þ Xn r¼1 o=ot E r þ 1 g r AeðtÞ ð2:23þ From Fig. 2.4 and (2.17) it is clear that the relaxation function for the generalized Maxwell model is, for a generic value of s Eðt sþ ¼E 1 þ Xn r¼1 E r e t s Tr ; T r ¼ g r =E r ð2:24þ The generalized Maxwell model provides an exponentially varying stress adding contributions with different relaxation times, one for each element in the chain. Thus, it is possible to fit experimental creep curves to any required degree of approximation if enough terms are used. To find the creep function for the generalized Maxwell model the differential Eq has to be solved, like in Example 1 below. The generalized Kelvin model is composed of n Kelvin units in series plus an isolated spring. The stress at each unit is the same external stress r(t) while the total (observable) strain e(t) is the sum of the internal strains in each element. Writing (2.19) in the symbolic form for a generic Kelvin element r r r ðtþ ¼ E r þ g o r =ot e r ðtþ ð2:25þ we have for the model in Fig. 2.4b 1 E 0 þ Xn r¼1! 1 rðtþ E r þ g r o=ot ð2:26þ

8 18 2 Rheological Models: Integral and Differential Representations From Eq and Fig. 2.4b, it is easy to gather that the specific creep function for the generalized Kelvin model is, for a generic value of s, Dðt sþ ¼ 1 E 0 þ Xn r¼1 1 E r ½1 e t s hr Š ; h r ¼ g r =E r ð2:27þ To find the relaxation function, the differential equation (2.26) has to be solved. Example 1 Determine the differential equation of the Zener model, that is a particular case of the generalized Maxwell model composed by a Maxwell model with parameters E 1 = E, g 1 = g in parallel with a spring of stiffness E?, Fig. 2.4a. Substituting these values into (2.23) we obtain! rðtþ ¼ E 1 þ o=ot eðtþ ð2:28þ Developing this symbolic equation we find o=ot E þ 1 g r þ g E _r ¼ E 1e þ gðe 1 þ EÞ _e E ð2:29þ With E z0 = E?? E, h z = g(e?? E)/(E? E) = ge z0 /[E? (E z0 - E? )] and T z = g/e = g/(e z0 - E? ), we have the nice form E z ð0þ _eðtþþ eðtþ ¼ _rðtþþ rðtþ ð2:30þ h z T z where E z (0) = E z0. Solving in e we obtain, with the initial condition e( ) = r( )/ E z0, rðtþ E z ð1þ 1 E z ð1þ 1 e ðt sþ hz _rðsþds ð2:31þ E z ð0þ 0 being E z (?) = E?. The corresponding creep function is then Dðt sþ ¼ 1 E z ð1þ 1 E zð0þ E z ð1þ e ðt sþ hz E z ð0þ ð2:32þ Example 2 Determine the differential equation of the standard solid model which is a particular case of the generalized Kelvin model with a spring (E 0 ) and a Kelvin element (E 1 = E, g 1 = g) connected in series. Substituting these parameters in (2.26), we have 1 1 þ rðtþ ð2:33þ E 0 E þ go=ot

9 2.3 Generalized Models 19 Developing this equation, the following differential equation is obtained r þ g E 0 þ E _r ¼ E 0E E 0 þ E e þ E 0g E 0 þ E _e ð2:34þ Making E s (0) = E 0, E s (?) = E 0 E/(E 0? E), h s = g/e and T s = g/(e 0? E), this differential equation can be written as E s ð0þ _eðtþþ eðtþ ¼ _rðtþþ rðtþ ð2:35þ h s T s Comparing (2.35) to(2.30), we conclude that the standard and Zener models present similar differential equations. Then, the solution for each one of these models can be obtained from the solution of the other by a convenient change of parameters General Differential Representation Equations 2.23 and 2.26 are differential equations with the general form X h i¼0 p i o i r ot i ¼ Xk j¼0 q j o j e ot j ð2:36þ where p i and q j are material constants dependent on the viscoelastic model. Usually, without loss of generality, we assume p 0 = 1. From (2.30), the constants for the Zener model are p 0 ¼ 1; p 1 ¼ g E ; q 0 ¼ E 1 and q 1 ¼ gðe 1 þ EÞ ð2:37þ E and, for the standard solid model (2.35), p 0 ¼ 1; p 1 ¼ g E 0 þ E ; q 0 ¼ E 0E E 0 þ E and q 1 ¼ E 0g E 0 þ E ð2:38þ Generalized Kelvin and Maxwell models are equivalent, in the sense that it is always possible to find a generalized Maxwell model equivalent to a given generalized Kelvin one, as in Examples 1 and 2. In Chap. 5 it is shown how to go from a creep to a relaxation function. Then, with Eqs. (2.24) and (2.27) we can find the corresponding models.

10 20 2 Rheological Models: Integral and Differential Representations 2.4 Integral and Differential Operators Viscoelastic relationships may also be indicated in the symbolic forms e ¼ D r r ¼ E e ð2:39þ which have to be interpreted as alternatives to (2.1) and (2.4) respectively. As the linear operators E* and D* may be handled formally as algebraic quantities (with some care), this notation simplifies some calculations. The operational form is valid also for the differential representation. For example, the differential operator for the generalized Kelvin model is the expression inside the brackets in (2.26). With this notation viscoelastic and elastic equations have similar form. A more rigorous development and applications of the operational technique will be given in Chap. 5 through the use of Laplace transform. Sometimes we need to invert the viscoelastic relations, i.e., to obtain the relaxation function corresponding to a given creep function and vice versa. From (2.39), we have Thus, e ¼ D E e ð2:40þ Hðt Þ¼D Eðt sþhðt Þ In extended form, this is written [4, 5] ð2:41þ 1 ¼ Dðt sþeðt tþþ Dðt sþ _Eðs Þds for t ð2:42þ Equation (2.42) express the obvious fact that applying as a stress history the corresponding relaxation function, we obtain a constant unit deformation. Example 3 Consider the relaxation function corresponding to the Zener model with E(t) = E 1? E 2 e -t/t ; then _EðtÞ ¼ E 2 e t=t =T and substituting into (2.42) Dðt ÞðE 1 þ E 2 Þ E 2 Dðt sþe ðt sþ=t ds ¼ 1 T Differentiating (2.43) in relation to t (Leibnitz rule) ð2:43þ _Dðt ÞðE 1 þ E 2 Þ E 2 T Dðt Þþ E 2 T 2 Dðt sþe ðt sþ=t ds ¼ 0 ð2:44þ Multiplying this equation by T and adding to (2.43) we eliminate the integral to obtain the differential equation ðe 1 þ E 2 ÞT _Dðt ÞþE 1 Dðt Þ¼1 ð2:45þ

11 2.4 Integral and Differential Operators 21 From which we obtain, with the initial condition D(0) = 1/E(0) = 1/(E 1? E 2 ) E 2 Dðt Þ¼ 1 e ðt sþ=h E 1 ðe 1 þ E 2 ÞE 1 ð2:46þ with h = (1? E 2 /E 1 )T. The operational form is valid also for the differential representation. For example, the differential operator for the generalized Kelvin model is the expression inside the brackets in (2.26). 2.5 Thermodynamic Restrictions The work done in deforming a viscoelastic body must be non-negative. Sufficient conditions that the relaxation function must satisfy are given in [6]. In reference to Eq. (2.5) (1) E(t) must be non-negative (2) E(t) must be a monotonically decreasing function with finite limit for t??. (3) E(t) must be convex downward. Most of the functions that are usually used to approximate the relaxation function satisfy the conditions above. References 1. F. Riesz, B. Sz.-Nagy, Functional Analysis (Dover Publications Inc., New York, 1990) 2. W. Flugge, Viscoelasticity (Springer, New York, 1975) 3. R.M. Christensen, Theory of Viscoelasticity, 2nd edn. (Dover Publications Inc, New York, 2010) 4. G.J. Creus, Viscoelasticity-Basic Theory and Application to Concrete Structures (Springer, Berlin, 1986) 5. A.C. Pipkin, Lectures on Viscoelasticity Theory (Springer, Heidelberg, 1972) 6. S. Breuer, E. Onat, On uniqueness in linear viscoelasticity. Q. Appl. Math. 19(4), pp (1962)

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